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Properties

Label 2-624-39.17-c0-0-0
Degree $2$
Conductor $624$
Sign $0.859 - 0.511i$
Analytic cond. $0.311416$
Root an. cond. $0.558047$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (1.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)13-s + 1.73i·21-s − 25-s + 0.999·27-s + 1.73i·31-s − 0.999·39-s + (−0.5 − 0.866i)43-s + (1 − 1.73i)49-s + (−0.5 − 0.866i)61-s + (−1.49 − 0.866i)63-s + (−1.5 − 0.866i)67-s − 1.73i·73-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (1.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)13-s + 1.73i·21-s − 25-s + 0.999·27-s + 1.73i·31-s − 0.999·39-s + (−0.5 − 0.866i)43-s + (1 − 1.73i)49-s + (−0.5 − 0.866i)61-s + (−1.49 − 0.866i)63-s + (−1.5 − 0.866i)67-s − 1.73i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :0),\ 0.859 - 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9024091790\)
\(L(\frac12)\) \(\approx\) \(0.9024091790\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + T^{2} \)
7 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - 1.73iT - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.83262064549757261779025819985, −10.29586929035861436136625478662, −9.176867286509257525104333490791, −8.390855695567502536544133074677, −7.37301608915810004823649888907, −6.32871874525778602190014383538, −5.15284744389746398276964585931, −4.46788908484202529413591837651, −3.57909551619564285295912477576, −1.59513767482707627453932159978, 1.50515612135362487738805585777, 2.61655853986043007549936872830, 4.41813693880371658290269596008, 5.54484731255588674475720137007, 5.97583254475485776225850655691, 7.39264452533451562382393925326, 8.065784939512042983425540800756, 8.668807895706193204494562136858, 9.998511309693193538274555644499, 11.16326648644933549402327645131

Graph of the $Z$-function along the critical line